Abelian extensions of Lie triple systems with derivations

Abstract

Let $ \mathfrak{L} $ and $ A $ be Lie triple systems, and let $ \theta_A $ be a representation of $ \mathfrak{L} $ on $ A. $ We first construct the third-order cohomology classes by derivations of $ A $ and $ \mathfrak{L}, $ then obtain a Lie algebra $ G_{\theta_A} $ with a representation $ \Phi $ on $ H^3(\mathfrak{L}, A), $ where $ \theta_A $ is given by an abelian extension $ 0\longrightarrow A\longrightarrow {\tilde {\mathfrak{L}}} \xrightarrow{\pi} \mathfrak{L}\longrightarrow 0. $ We study obstruction classes for extensibility of derivations of $ A $ and $ \mathfrak{L} $ to those of $ \tilde{\mathfrak{L}}. $ An application of $ \Phi $ is discussed.

Keywords

• lie triple system
• cohomology
• derivation
• abelian extension